Herbert A. Hauptman

Structure solution by minimal-function phase refinement and Fourier filtering. II. Implementation and applications

The minimal function, R(psi), has been used to provide the basis for a new computer-intensive direct-methods procedure that shows potential for providing fully automatic routine solutions for structures in the 200-400 atom range. This procedure, which has been called shake-and-bake, is an iterative process in which real-space filtering is alternated with phase refinement using a technique that reduces the value of R(psi). It has been successfully tested using experimental data for a dozen known structures ranging in size from 25 to 317 atoms and crystallizing in a variety of space groups. The details of this procedure, the parameters used and the results of these applications are described.

Valinomycin Crystal Structure Determination by Direct Methods

The conformation of an uncomplexed form of the antibiotic valinomycin (C54N6O18H90) has been determined by direct methods including a novel technique for strong enantiomorph discrimination via the calculation and systematic analysis of cosine invariants of a special type. The intramolecular hydrogen bonding scheme and the isopropyl group stereochemistry of uncomplexed valinomycin are compatible with interpretations of spectral measurements for the complexed and uncomplexed molecule in solution but are different from any previously proposed structure. The simple conformational change of a hydrogen bond shift, which could be induced by the process of potassium ion complexing, transforms the uncomplexed into the complexed structure.

Structure solution by minimal-function phase refinement and Fourier filtering. I. Theoretical basis

Eliminating the N atomic position vectors rj, j = 1, 2, ..., N, from the system of equations defining the normalized structure factors EH yields a system of identities that the EHs must satisfy, provided that the set of EHs is sufficiently large. Clearly, for fixed N and specified space group, this system of identities depends only on the set [H], consisting of n reciprocal-lattice vectors H, and is independent of the crystal structure, which is assumed for simplicity to consist of N identical atoms per unit cell. However, for a fixed crystal structure, the magnitudes magnitude of /EH/ are uniquely determined so that a system of identities is obtained among the corresponding phases psi H alone, which depends on the presumed known magnitudes magnitude of /EH/ and which must of necessity be satisfied. The known conditional probability distributions of triplets and quartets, given the values of certain magnitudes magnitude of /E/, lead to a function R(psi) of phases, uniquely determined by magnitudes magnitude of /E/ and having the property that RT < 1/2 < RR, where RT is the value of R(psi) when the phases are equal to their true values, no matter what the choice of origin and enantiomorph, and RR is the value of R(psi) when the phases are chosen at random. The following conjecture is therefore plausible: the global minimum of R(psi), where the phases are constrained to satisfy all identities among them that are known to exist, is attained when the phases are equal to their true values and is thus equal to RT.(ABSTRACT TRUNCATED AT 250 WORDS)

Use of negative quartet cosine invariants as a phasing figure of merit: NQEST

Recent theoretical advances in the identification of those cosine invariants cos (ϕh + ϕk + ϕ1 + ϕm) which are probably negative suggest algorithms for the calculation of a figure of merit which is sensitive to the integrity of a phase set. The negative quartet figure of merit, NQEST, defined here is of particular utility in conjunction with fast multi-solution tangent formula techniques. Development of the methods and applications to both known and unknown crystal structures are presented.

The Direct Methods of X-ray Crystallography

The electron density function ρ(r) in a crystal determines its difaction pattern, that is, both the magnitudes and phases of its x-ray diffraction maxima, and conversely. If, however, as is always the case, only magnitudes are available from the diffraction experiment, then the density function ρ(r) cannot be recovered. If one invokes prior structural knowledge, usually that the crystal is composed of discrete atoms of known atomic numbers, then the observed magnitudes are, in general, sufficient to determine the positions of the atoms, that is, the crystal structure.

Ab initio structure determination and refinement of a scorpion protein toxin.

The structure of toxin II from the scorpion Androctonus australis Hector has been determined ab initio by direct methods using SnB at 0.96 A resolution. For the purpose of this structure redetermination, undertaken as a test of the minimal function and the SnB program, the identity and sequence of the protein was withheld from part of the research team. A single solution obtained from 1 619 random atom trials was clearly revealed by the bimodal distribution of the final value of the minimal function associated with each individual trial. Five peptide fragments were identified from a conservative analysis of the initial E-map, and following several refinement cycles with X-PLOR, a model was built of the complete structure. At the end of the X-PLOR refinement, the sequence was compared with the published sequence and 57 of the 64 residues had been correctly identified. Two errors in sequence resulted from side chains with similar size while the rest of the errors were a result of severe disorder or high thermal motion in the side chains. Given the amino-acid sequence, it is estimated that the initial E-map could have produced a model containing 99% of all main-chain and 81% of side-chain atoms. The structure refinement was completed with PROFFT, including the contributions of protein H atoms, and converged at a residual of 0.158 for 30 609 data with F >or= 2sigma(F) in the resolution range 8.0-0.964 A. The final model consisted of 518 non-H protein atoms (36 disordered), 407 H atoms, and 129 water molecules (43 with occupancies less than unity). This total of 647 non-H atoms represents the largest light-atom structure solved to date.

Application of the minimal principle to peptide structures.

A new direct-methods procedure has been devised which consists of phase refinement via the minimal function, R(phi), alternated with Fourier summation and real space filtering. All phases are initially assigned values by computing structure factors for a randomly positioned set of atoms. These phases are then refined by using a parameter shift method to minimize R(phi). The refined phases are Fourier transformed, and a specified number of the largest peaks in the electron-density function are found and used as a new trial structure. The probability of a trial structure converging to a solution appears to depend on structural complexity and a number of refinement parameters. This procedure shows potential for providing fully automatic routine solutions for structures in the 200-400 atom range.

The Phase Problem of X‐Ray Crystallography

With the invention some 200 years ago of the goniometer, an instrument for measuring the angles between the faces of a crystal, the science of crystallography was born. The goniometer made possible the discovery of the fundamental laws of descriptive crystallography: that the angles between the facial planes are determined by the chemical composition of the crystal and that the relative orientations of the facial planes follow a simple rule, the law of rational positions.

Incorporating tangent refinement in the Shake-and-Bake formalism

Shake-and-Bake is a direct-methods procedure in which phase refinement and Fourier refinement are alternated repetitively, unconditionally and automatically. The traditional Shake-and-Bake approach invoked a parameter-shift routine to perform phase refinement in an effort to reduce the value of minimal function. In this paper, parameter shift is replaced with the tangent formula as a means of phase refinement. This study shows that the tangent formula is more efficient than parameter shift for small structures when the number of refinement cycles and number of applications of the tangent formula per Shake-and-Bake cycle are chosen very carefully. For larger structures, including the 400 non-H-atom crambin structure, the two methods generally perform with similar efficiency. However, only parameter shift has successfully produced recognizable solutions for the difficult 317 non-H-atom structure gramicidin A.

Exponential Shake-and-Bake: theoretical basis and applications.

The simple cosine function used in the formulation of the traditional minimal principle and the related Shake-and-Bake algorithm is here replaced by a function of exponential type and its expected value and variance are derived. These lead to the corresponding exponential minimal principle and its associated Exponential Shake-and-Bake algorithm. Recent applications of the exponential function to several protein structures within the Shake-and-Bake framework suggest that this function leads, in general, to significant improvements in the success rate (percentage of trial structures yielding solution) of the Shake-and-Bake procedure. However, only in space group P1 is it presently possible to assign optimal values a priori for the exponential-function parameters.